Contemporary
engineering design is heavily based on computer
simulation. Traditional optimization techniques directly
utilize the simulated responses and possibly available derivatives to
make the responses to satisfy given design specifications.

In
many cases, the computational cost of such an optimization process may
be prohibitive, especially for complex problems. This is not only
because of the long simulation time for a single design, which,
depending on the system complexity and required accuracy, may be as
long as several hours or even a couple of days, but also because the
number of simulations required to obtain convergence of the
optimization algorithm is usually large (e.g., hundreds) for
traditional direct methods.

On the other hand, sensitivity
information is normally unavailable, whereas sensitivity estimation
using, e.g., finite differences, may be inaccurate due to various
reasons, such as discontinuity of the system response as a function of
the design parameters, related to re-meshing the structure inside the
simulator whenever the design variables are being modified.
Even in cases, when using finite differences is numerically feasible,
it is usually computationally too expensive.

Another problem
is that objective functions may be quite nonlinear and very often one
has to face multiple local optima, which normally requires restarting
of the optimization process with different (e.g., randomly selected)
initial designs. This, obviously, increases the computational cost of
the optimization process.

All of these problems are especially
visible in microwave engineering, one of the focus areas of EOMC, because microwave structures are
normally evaluated through accurate but CPU intensive full-wave
electromagnetic simulations.

Surrogate-Based Optimization for Microwave Engineering

One of EOMC's endeavors
is the development of computationally-efficient design
optimization algorithms for microwave/RF engineering. Of particular
interest are the methods exploiting so-called surrogate-based
optimization (SBO) principle [1]. The main idea is that the direct
optimization of the microwave structure with a CPU-intensive objective
function is replaced by the iterative updating and re-optimization of a
so-called surrogate model. The surrogate model is a simplified but
computationally cheap and analytically tractable representation of the
original structure subsequently called the fine model.

Space Mapping

The
most
successfull SBO approach in microwave engineering is space mapping
(SM) [2], [3]. According to SM, the surrogate model is constructed
using
a physically-based and computationally cheap representation of the
structure being optimized, so-called coarse model (typically, circuit
equivalent), and auxiliary (usually linear) mappings. Some introductory
material regarding space mapping as well as review of recent
developments can be found in Koziel et al. 2008,Koziel et al. 2006 and Bandler et al. 2006.

EOMC is working on all aspects of space mapping optimization,
including theory, algorithm development, robustness, convergence,
surrogate model selection, and software implementation.A well
performing SBO procedure, in particular, space maping algorithm,
requires only a few evaluations of a CPU-intensive structure being
optimized to yield a satisfactory design. Most of the optimization
burden is shifted to a computationally cheap surrogate model.

Space
mapping algorithm requires four iterations (i.e., five electromagnetic
simulations of the filter structure) to find the optimal design. The
plot below shows the fine model responses corresponding to the
initial design (dashed line) and the final design (solid line):

Space Mapping with Functional Approximation

Normally,
space mapping requires fast coarse model (typically, equivalent
circuit), however, EOMC works on alternative formulations of the
technique that allows us to use SM even in cases when the
computationally cheap coarse model is not available (e.g., antennas or
waveguide structures). One possible way, explored in EOMC, is to build
a coarse model using coarse-mesh EM simulations and available classical
function approximation techniques such as kriging.

Illustration:
The use of functional and physical surrogate modeling in microwave CAD
can be illustrated using the example of the microstrip fed monopole
antenna:

The fine model of the antenna is evaluated in CST Microwave Studio
(simulation time about 2.5 hours). The design specifications are |S11|
≤ –10 dB for 3.1 GHz to 10.6 GHz. We want to optimize the antenna using
space mapping, however, no equivalent-circuit coarse model is available
in this case. Instead, we use a coarse-discretization CST model
(evaluation time 2 minutes and 15 seconds). This model is still
computationally too expensive to be used directly as a coarse model in
the SM optimization process. Therefore, the coarse model is created in
the neighbourhood of the starting point (here, the approximate optimum
of the coarse-discretization model), using kriging interpolation of the
coarse-discretization model data. The coarse model created this
way is computationally cheap, easy to optimize, and yet retains the
features of a physically-based model.

The
plot below shows the fine (dotted line) and coarse (x) model responses
at the initial design, as well as fine (solid line) and coarse (dashed
line) model responses at the coarse model optimum. The next plot shows
the fine model response at the final design obtained after three SM
iterations with kriging-based surrogate model. The optimization
cost is 245 evaluations of the coarse-discretization model (145 to get
its optimized design and another 100 to set up the kriging surrogate)
as well as 4 fine model evaluations (including evaluation at the
initial design). Thus, the total cost corresponds to only 8 evaluations
of the fine model!

Tuning Space Mapping

Tuning
space mapping (TSM) is one of the latest developments in space mapping
technology. TSM algorithms offer a remarkably fast design optimization
with satisfactory results obtained after one or two iterations which
amounts to just a few electromagnetic simulations of the optimized
microwave structure. According to the TSM approach, the surrogate
model’s role is taken by a so-called tuning model, which is constructed
by introducing circuit-theory based components (e.g., capacitors,
inductors or coupled-line models) into the fine model structure, and
parameters of these circuit components are chosen to be tunable. The
tuning model is updated and optimized with respect to the tuning
parameters. With the optimal tuning parameters thus obtained, a
calibration is needed to transform these tuning values into an
appropriate modification of the design variables, which are then
assigned to the fine model. The calibration process may involve
analytical formulas or it may require an auxiliary model, typically a
fast space mapping surrogate.

EOMC is working on various
aspects of TSM, including automated implementation of TSM algorithms
where all the interactions between various models involved in the
optimization process is handled by the SMF system and does not require user intervention.

Illustration: Design optimization of the box-section Chebyshev microstrip bandpass filter. The fine model is simulated in Sonnet em. The tuning model is constructed by dividing the polygons corresponding to parameters L1 to L5 in the middle and inserting the tuning ports at the new cut edges as shown below:

Its
S28P data file is then loaded into the S-parameter component in Agilent
ADS. The circuit-theory coupled-line components and capacitor
components are chosen to be the tuning elements and are inserted into
each pair of tuning ports. The lengths of the imposed coupled-lines and
the capacitances of the capacitors are assigned to be the tuning
parameters:

The
calibration model is implemented in ADS. It contains the same tuning
elements as the tuning model. It basically mimics the division of the
coupled-lines performed while preparing the tuning model. The
calibration model also contains six (implicit) SM parameters that are
be used in the calibration process:

The
plots below show the coarse (dashed line) and fine (solid line) model
response at the initial design, as well as the fine model response
after just one TSM iteration.

Shape Preserving Response Prediction

EOMC
is constantly working towards improving efficiency of simulation-based
design optimization process. One of the recent developments is
shape-preserving response prediction (SPRP). SPRP relies on a
physically-based coarse model. The enhanced coarse model is a surrogate
that is being optimized instead of the fine model. The surrogate
coincides with the fine model at the starting point of any given
iteration, and the surrogate model response change is generated based
on the translation vectors of a set of characteristic points of the
coarse model response. Because of these features, the surrogate model
exhibits very good prediction capability and, therefore, permits
efficient optimization of the fine model.

The SPRP concept is explained in the pictures below. Picture (a) shows the example coarse model response at the reference design x0 (solid line),
the coarse model response at other design x (dotted line),
characteristic points of both responses (circles and squares), and the
translation vectors (short lines). Picture (b) shows the fine model response at x0
(solid line) and the predicted fine model response at x (dotted line)
obtained using SPRP based on characteristic points of picture (a);
characteristic points of the fine model response at x0 (circles) and
the translation vectors (short lines) were used to find the
characteristic points (squares) of the predicted fine model response;
coarse model responses at x0 and at x are plotted using thin solid
and dotted line, respectively.

(a)

(b)

Illustration: Optimization of the wideband bandstop filter. The fine model is simulated in FEKO:

The coarse model is implemented in Agilent ADS:

Six characteristic points are selected to set up SPRP surrogate model: two points for which |S21| = –3 dB, two points with |S21| = –20 dB, and the two local |S21|
maxima. SPRP optimization process needs only three iterations (i.e.,
three EM simulation of the structure) to yield a very good design. The
plot below shows the fine model (dashed line) and coarse model
(thin dashed line) response at the initial design, and the optimized
fine model response (solid line):

EOMC Focus

The
algorithms and optimization methods developed in EOMC are exploiting
SBO principles, in particular space mapping, as well as related
technologies such as tuning [5] and adaptive response correction [6].EOMC
addresses all aspects of computationally efficient design
optimization including the theory, the algorithm development,
and the applications to EM-based
design and optimization of RF/microwave components and circuits, as
well as to design problems in other fields, particularly, aerospace
engineering.EOMC
also works on the development of the user-friendly software
implementing the state-of-the-art design optimization algorithms.
Special emphasis is put on interfacing commercial EM/circuit
simulators, which is a key step in making the optimization process
fully automatic.